Problem: Simplify $\dfrac{5+12i}{2-3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary).
Explanation: Multiplying the numerator and denominator by the conjugate of the denominator, we have \begin{align*}
\dfrac{5+12i}{2-3i} \cdot \frac{2+3i}{2+3i} &= \frac{5(2) + 5(3i) + 12i(2) +12i(3i)}{2(2) + 2(3i) + -3i(2) -3i(3i)}\\
& = \frac{-26+39i}{13} \\
&= \boxed{-2+3i}.
\end{align*}